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 yidongxia August 4, 2012 21:26

Gradient evaluation in Finite Volume Methods

Is there a method that is based on "coordinate transformation" to evaluate the gradient of the variables at the interface between two adjacent cells in 2D provided the coordinates of the two cell centers and two points that link the interface line? If you happen to know that, would you give me a hint? :D Thanks very much!

 leflix August 5, 2012 12:26

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Quote:
 Originally Posted by yidongxia (Post 375409) Is there a method that is based on "coordinate transformation" to evaluate the gradient of the variables at the interface between two adjacent cells in 2D provided the coordinates of the two cell centers and two points that link the interface line? If you happen to know that, would you give me a hint? :D Thanks very much!

Hi Yidonga,

if you use finite volume,you do not need to use coordinate transformation.
This should answer to you question

leflix,
would you please introduce the reference you got the method?

 yidongxia August 5, 2012 16:04

Thanks a lot! This question originally comes from a teacher's assignment in FV CFD, and I really have no idea what it means by "based on coordinate transformation theory", since from my knowledge, "coordinate transformation" is a concept from Finite Element Method, but not Finite Volume Method. Except from the method you provide, I also know there is the Green-Gauss method and least-squares method. Do you know any other method that can work good? Thanks!:D

Quote:
 Originally Posted by leflix (Post 375463) Hi Yidonga, if you use finite volume,you do not need to use coordinate transformation. This should answer to you question

 leflix August 5, 2012 18:54

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Quote:
 Originally Posted by hadian (Post 375465) leflix, would you please introduce the reference you got the method?

the method is just derivation of simple differential geometry and vector calculus.
check the pdf file.

Journal of Computational Physics 162, 411–428 (2000)
Computers & Fluids 57, 225–236 (2012)
Journal of Computational Physics , 228:5148–59 (2009)

you have different implementations possible for this normal gradient evaluation. one which involves phi_a and phi_b and requires additional interpolation process but which does not require gradient reconstruction (ref 1 and 2). And one which avoids theses interpolations but requires gradient reconstruction.

 leflix August 5, 2012 19:15

Quote:
 Originally Posted by yidongxia (Post 375478) I really have no idea what it means by "based on coordinate transformation theory", since from my knowledge, "coordinate transformation" is a concept from Finite Element Method, but not Finite Volume Method.
I think you misunderstood something. Coordinate transform "method" consists in transforming a physical domain meshed with boundary fiitted curvilinear coordinate into a computational cartesian orthogonal grid. Then you can use classical FDM. Once you got the solution in the cartesian grid, a reverse transformation gives you the solution in the physical domain. Sothis concept is really attached to FDM.
With FEM or FVM such transformation is absolutely not necessary.

Quote:
 Except from the method you provide, I also know there is the Green-Gauss method and least-squares method. Do you know any other method that can work good? Thanks!:D
Green-Gauss method or least-square method are gradient reconstruction methods. These methods aim to express GRAD(PHY) at all cell centers which is different than expressing GRAD(PHY)_e. n= dphy/dn
In the expression I provided you do not need to reconstruct the gradient. But you need to interpolate phy_a and phy_b from cell centers values. Different implementation would avoid these interpolation but will require gradient reconstruction.
See my previous post and check the book from Peric Computational Methods Fluid Dynamics.

other methods to reconstruct the gradient are based on shape functions. This time it is more linked with FEM. Knowing the value of your function at given nodes (typically the vertices of your cell), you try to determine an analytical function which will interpolate these values. Then with this analytical function is is easy to derive it and to obtain the gradient on the location you need.

 yidongxia August 6, 2012 10:11

Again, thanks so much for the detailed explanation! Now I'm clear that the requirement is to transform the physical domain (x,y) into the generalized coordinate system (xi,eta) and then compute gradient(U)·n x Area at interface. But I'm not sure how to write the transformation function
x=x(xi,eta) and y=y(xi,eta)
or
xi=(x,y) and eta=(x,y)
If I can know them, I think I'll be completely clear. Thank you!:D

Quote:
 Originally Posted by leflix (Post 375463) Hi Yidonga, if you use finite volume,you do not need to use coordinate transformation. This should answer to you question

 leflix August 6, 2012 10:23

Quote:
 Originally Posted by yidongxia (Post 375607) Again, thanks so much for the detailed explanation! Now I'm clear that the requirement is to transform the physical domain (x,y) into the generalized coordinate system (xi,eta) and then compute gradient(U)·n x Area at interface. But I'm not sure how to write the transformation function x=x(xi,eta) and y=y(xi,eta) or xi=(x,y) and eta=(x,y) If I can know them, I think I'll be completely clear. Thank you!:D
Here is a usefull reference for coordinate transform
http://www.erc.msstate.edu/publications/gridbook/

but as I stated it, if you use finite volume,you do not need to use such transformation. You directly work in the physical domain which is the same as computational domain.

or if you absolutely want to use coordinate transformation switch to finite difference method.

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